In mathematics, Krener's theorem is a result attributed to Arthur J. Krener in geometric control theory about the topological properties of attainable sets of finite-dimensional control systems.
It states that any attainable set of a bracket-generating system has nonempty interior or, equivalently, that any attainable set has nonempty interior in the topology of the corresponding orbit.
Heuristically, Krener's theorem prohibits attainable sets from being hairy.
be a smooth control system, where
belongs to a finite-dimensional manifold
belongs to a control set
Consider the family of vector fields
be the Lie algebra generated by
with respect to the Lie bracket of vector fields.
, if the vector space
belongs to the closure of the interior of the attainable set from
, the attainable set from
has nonempty interior in the orbit topology, as it follows from Krener's theorem applied to the control system restricted to the orbit through
When all the vector fields in
belongs to the closure of the interior of the attainable set from
This is a consequence of Krener's theorem and of the orbit theorem.
As a corollary of Krener's theorem one can prove that if the system is bracket-generating and if the attainable set from
, then the attainable set from